September 2019, 18(5): 2377-2395. doi: 10.3934/cpaa.2019107

Molecular characterization of anisotropic weak Musielak-Orlicz Hardy spaces and their applications

College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China

* Corresponding author

Received  April 2018 Revised  October 2018 Published  April 2019

Fund Project: The third author is supported by National Natural Science Foundation of China 11461065 & 11861062 & 11661075

Let $ A $ be a real $ n\times n $ matrix with all its eigenvalues $ \lambda $ satisfy $ |\lambda|>1 $. Let $ \varphi: \mathbb{R}^n\times[0, \, \infty)\to[0, \, \infty) $ be an anisotropic Musielak-Orlicz function, i.e., $ \varphi(x, \cdot) $ is an Orlicz function uniformly in $ x\in{\mathbb{R}^n} $ and $ \varphi(\cdot, \, t) $ is an anisotropic Muckenhoupt $ \mathcal {A}_\infty({\mathbb{R}^n}) $ weight uniformly in $ t\in(0, \, \infty) $. In this article, the authors introduce the anisotropic weak Musielak-Orlicz Hardy space $ WH^{\varphi}_A(\mathbb{R}^n) $ via the grand maximal function and establish its molecular characterization which are anisotropic extensions of Liang, Yang and Jiang (Math. Nachr. 289: 634-677, 2016). As an application, the boundedness of anisotropic Calderón-Zygmund operators from $ H_A^\varphi(\mathbb{R}^n) $ to $ WH_A^\varphi(\mathbb{R}^n) $ in the critical case is presented.

Citation: Ruirui Sun, Jinxia Li, Baode Li. Molecular characterization of anisotropic weak Musielak-Orlicz Hardy spaces and their applications. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2377-2395. doi: 10.3934/cpaa.2019107
References:
[1]

J. Álvarez, Hp and weak Hp continuity of Calderón-Zygmund type operators, in Fourier Analysis (Orono, ME, 1992), Lecture Notes in Pure and Appl. Math., 157 (1994), Dekker, New York, 17–34.

[2]

J. Álvarez, Continuity properties for linear commutators of Calderón-Zygmund operators, Collect. Math., 49 (1998), 17-31.

[3]

Z. Birnbaum and W. Orlicz, Über die verallgemeinerung des begriffes der zueinander konjugierten potenzen (German), Studia Math., 3 (1931), 1-67.

[4]

A. BonamiJ. Feuto and S. Grellier, Endpoint for the DIV-CURL lemma in Hardy spaces, Publ. Mat., 54 (2010), 341-358. doi: 10.5565/PUBLMAT_54210_03.

[5]

M. Bownik, Anisotropic Hardy spaces and wavelets, Mem. Amer. Math. Soc., 164 (2003), ⅵ+122 pp. doi: 10.1090/memo/0781.

[6]

M. Bownik and K.-P. Ho, Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces, Tran. Amer. Math. Soc., 358 (2006), 1469-1510. doi: 10.1090/S0002-9947-05-03660-3.

[7]

M. BownikB. LiD. Yang and Y. Zhou, Weighted anisotropic Hardy spaces and their applications in boundedness of sublinear operators, Indiana Univ. Math. J., 57 (2008), 3065-3100. doi: 10.1512/iumj.2008.57.3414.

[8]

M. BownikB. LiD. Yang and Y. Zhou, Weighted anisotropic product Hardy spaces and boundedness of sublinear operators, Math. Nachr., 283 (2010), 392-442. doi: 10.1002/mana.200910078.

[9]

T. BuiJ. CaoL. D. KyD. Yang and S. Yang, Musielak-Orlicz Hardy spaces associated with operators satisfying reinforced off-diagonal estimates, Anal. Geom. Metr. Spaces, 1 (2013), 69-129. doi: 10.2478/agms-2012-0006.

[10]

A. P. Calderón and A. Torchinsky, Parabolic maximal function associated with a distribution, Adv. Math., 16 (1975), 1-64. doi: 10.1016/0001-8708(75)90099-7.

[11]

J. CaoD.-C. ChangD. Yang and S. Yang, Boundedness of second order Riesz transforms associated to Schrödinger operators on Musielak-Orlicz-Hardy spaces, Commun. Pure Appl. Anal., 13 (2014), 1435-1463. doi: 10.3934/cpaa.2014.13.1435.

[12]

W. Chen and Y. Lai, Boundedness of fractional integrals in Hardy spaces with non-doubling measure, Anal. Theory Appl., 22 (2006), 195-200. doi: 10.1007/BF03218712.

[13]

R. R. Coifman and G. Weiss, Analyse Harmonique Non-commutative sur Certains Espaces Homogènes (French) [Non-commutative harmonic analysis on certain homogeneous spaces] Lecture Notes in Math., 242, Springer, Berlin, 1971.

[14]

L. Diening, Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces, Bull. Sci. Math., 129 (2005), 657–700. doi: 10.1016/j.bulsci.2003.10.003.

[15]

Y. Ding and S. Lan, Anisotropic weak Hardy spaces and interpolation theorems, Sci. China Math., 51 (2008), 1690-1704. doi: 10.1007/s11425-008-0009-z.

[16]

Y. Ding and S. Lan, Anisotropic Hardy estimates for multilinear operators, Adv. Math., 38 (2009), 168-184.

[17]

Y. Ding and S. Lu, Hardy spaces estimates for multilinear operators with homogeneous kernels, Nagoya Math. J., 170 (2003), 117–133. doi: 10.1017/S0027763000008552.

[18]

Y. DingS. Lu and S. Shao, Integral operators with variable kernels on weak Hardy spaces, J. Math. Anal. Appl., 317 (2006), 127-135. doi: 10.1016/j.jmaa.2005.10.085.

[19]

R. Fefferman and F. Soria, The spaces weak H1, Studia Math., 85 (1986), 1-16. doi: 10.4064/sm-85-1-1-16.

[20]

J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam, 1985.

[21]

L. Grafakos, Hardy space estimates for multilinear operators, Ⅱ, Rev. Mat. Iberoam., 8 (1992), 69-92. doi: 10.4171/RMI/117.

[22]

L. Grafakos, Modern Fourier Analysis, 2ed edition, Graduate Texts in Mathematics 250, Springer, New York, 2009. doi: 10.1007/978-0-387-09434-2.

[23]

S. Hou, D. Yang and S. Yang, Lusin area function and molecular characterizations of MusielakOrlicz Hardy spaces and their applications, Commun. Contemp. Math., 15 (2013), 1350029, 37 pp. doi: 10.1142/S0219199713500296.

[24]

R. Johnson and C. J. Neugebauer, Homeomorphisms preserving Ap, Rev. Mat. Iberoam., 3 (1987), 249-273. doi: 10.4171/RMI/50.

[25]

N. J. Kalton, Linear operators on Lp for 0 < p < 1, Trans. Amer. Math. Soc., 259 (1980), 319-355. doi: 10.2307/1998234.

[26]

L. D. Ky, Bilinear decompositions and commutators of singular integral operators, Trans. Amer. Math. Soc., 365 (2013), 2931-2958. doi: 10.1090/S0002-9947-2012-05727-8.

[27]

L. D. Ky, New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators, Integral Equations Operator Theory, 78 (2014), 115-150. doi: 10.1007/s00020-013-2111-z.

[28]

L. D. Ky, Bilinear decompositions for the product space $H^1_L\times BMO_L$, Math. Nachr., 287 (2014), 1288–1297. doi: 10.1002/mana.201200101.

[29]

B. LiX. Fan and D. Yang, Littlewood-Paley theory of anisotropic Hardy spaces of Musielak-Orlicz type, Taiwanese J. Math., 19 (2015), 279-314. doi: 10.11650/tjm.19.2015.4692.

[30]

B. LiX. FanZ. Fu and D. Yang, Molecular characterization of anisotropic Musielak-Orlicz Hardy spaces and their applications, Acta Math. Sin., 32 (2016), 1391-1414. doi: 10.1007/s10114-016-4741-y.

[31]

B. Li, R. Sun, M. Liao and B. Li, Littlewood-Paley characterizations of anisotropic weak Musielak-Orlicz Hardy spaces, Nagoya Math. J., 1–40. doi: 10.1017/nmj.2018.10.

[32]

B. Li, D. Yang and W. Yuan, Anisotropic Hardy spaces of Musielak-Orlicz type with applications to boundedness of sublinear operators, The Scientific World Journal, (2014), 19 pp.

[33]

Y. LiangJ. Huang and D. Yang, New real-variable characterizations of Musielak-Orlicz Hardy spaces, J. Math. Anal. Appl., 395 (2012), 413-428. doi: 10.1016/j.jmaa.2012.05.049.

[34]

Y. LiangD. Yang and R. Jiang, Weak Musielak-Orlicz Hardy spaces and applications, Math. Nachr., 289 (2016), 637-677. doi: 10.1002/mana.201500152.

[35]

H. Liu, The weak Hp spaces on homogeneous groups, in Harminic Analysis (Tianjin, 1998), Lecture Notes in Math. Vol., 1984 (Springer, Berlin, 1991), pp. 113–118. doi: 10.1007/BFb0087762.

[36]

J. Liu, F. Weisz, D. Yang and W. Yuan, Littlewood-Paley and finite atomic characterizations of anisotropic variable Hardy-Lorentz spaces and their applications, J. Fourier Anal. Appl., (2018). doi: 10.1007/s00041-018-9609-3.

[37]

J. LiuD. Yang and W. Yuan, Anisotropic Hardy-Lorentz spaces and their applications, Sci. China Math., 59 (2016), 1669-1720. doi: 10.1007/s11425-016-5157-y.

[38]

J. LiuD. Yang and W. Yuan, Anisotropic variable Hardy-Lorentz spaces and their real interpolation, J. Math. Anal. Appl., 456 (2017), 356-393. doi: 10.1016/j.jmaa.2017.07.003.

[39]

J. Musielak, Orlicz Spaces and Modular Spaces, Springer-Verlag, Berlin-NewYork, 1983. doi: 10.1007/BFb0072210.

[40]

W. Orlicz, Über eine gewisse Klasse von Raumen vom Typus B (German), Bull. Int. Acad. Pol. Ser. A, 8 (1932), 207-220.

[41]

T. Quek and D. Yang, Calderón-Zygmund-type operators on weighted weak Hardy spaces over ${{\mathbb R}^n}$, Acta Math. Sin., 16 (2000), 141-160. doi: 10.1007/s101149900022.

[42]

E. M. SteinM. Taibleson and G. Weiss, Weak type estimates for maximal operators on certain Hp classes, Rend. Circ. Mat. Palermo, 1 (1981), 81-97.

[43]

J.-O. Strömberg and A. Torchinsky, Weighted Hardy Spaces, Berlin-New York: Springer-Verlag, 1989. doi: 10.1007/BFb0091154.

[44]

M. H. Taibleson and G. Weiss, The molecular characterization of certain Hardy spaces, Asterisque, Mathematical Reviews, 77 (1980), 67–149.

[45]

L. Tang, The weighted weak local Hardy spaces, Rocky Mountain J. Math., 44 (2014), 297-315. doi: 10.1216/RMJ-2014-44-1-297.

[46]

H. Wang, Boundedness of several integral operators with bounded variable kernels on Hardy and weak Hardy spaces, Internat. J. Math., 24 (2013), 1350095, 22 pp. doi: 10.1142/S0129167X1350095X.

[47]

D. Yang and S. Yang, Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated to non-negative self-adjoint operators satisfying Gaussian estimates, Commun. Pure Appl. Anal., 15 (2016), 2135-2160. doi: 10.3934/cpaa.2016031.

[48]

D. Yang, Y. Liang and L. D. Ky, Real-Variable Theory of Musielak-Orlicz Hardy Spaces, Lecture Notes in Mathematics, 2182, Springer-Verlag, Cham, 2017. doi: 10.1007/978-3-319-54361-1.

[49]

H. ZhangC. Qi and B. Li, Anisotropic weak Hardy spaces of Musielak-Orlicz type and their applications, Front. Math. China, 12 (2017), 993-1022. doi: 10.1007/s11464-016-0546-7.

show all references

References:
[1]

J. Álvarez, Hp and weak Hp continuity of Calderón-Zygmund type operators, in Fourier Analysis (Orono, ME, 1992), Lecture Notes in Pure and Appl. Math., 157 (1994), Dekker, New York, 17–34.

[2]

J. Álvarez, Continuity properties for linear commutators of Calderón-Zygmund operators, Collect. Math., 49 (1998), 17-31.

[3]

Z. Birnbaum and W. Orlicz, Über die verallgemeinerung des begriffes der zueinander konjugierten potenzen (German), Studia Math., 3 (1931), 1-67.

[4]

A. BonamiJ. Feuto and S. Grellier, Endpoint for the DIV-CURL lemma in Hardy spaces, Publ. Mat., 54 (2010), 341-358. doi: 10.5565/PUBLMAT_54210_03.

[5]

M. Bownik, Anisotropic Hardy spaces and wavelets, Mem. Amer. Math. Soc., 164 (2003), ⅵ+122 pp. doi: 10.1090/memo/0781.

[6]

M. Bownik and K.-P. Ho, Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces, Tran. Amer. Math. Soc., 358 (2006), 1469-1510. doi: 10.1090/S0002-9947-05-03660-3.

[7]

M. BownikB. LiD. Yang and Y. Zhou, Weighted anisotropic Hardy spaces and their applications in boundedness of sublinear operators, Indiana Univ. Math. J., 57 (2008), 3065-3100. doi: 10.1512/iumj.2008.57.3414.

[8]

M. BownikB. LiD. Yang and Y. Zhou, Weighted anisotropic product Hardy spaces and boundedness of sublinear operators, Math. Nachr., 283 (2010), 392-442. doi: 10.1002/mana.200910078.

[9]

T. BuiJ. CaoL. D. KyD. Yang and S. Yang, Musielak-Orlicz Hardy spaces associated with operators satisfying reinforced off-diagonal estimates, Anal. Geom. Metr. Spaces, 1 (2013), 69-129. doi: 10.2478/agms-2012-0006.

[10]

A. P. Calderón and A. Torchinsky, Parabolic maximal function associated with a distribution, Adv. Math., 16 (1975), 1-64. doi: 10.1016/0001-8708(75)90099-7.

[11]

J. CaoD.-C. ChangD. Yang and S. Yang, Boundedness of second order Riesz transforms associated to Schrödinger operators on Musielak-Orlicz-Hardy spaces, Commun. Pure Appl. Anal., 13 (2014), 1435-1463. doi: 10.3934/cpaa.2014.13.1435.

[12]

W. Chen and Y. Lai, Boundedness of fractional integrals in Hardy spaces with non-doubling measure, Anal. Theory Appl., 22 (2006), 195-200. doi: 10.1007/BF03218712.

[13]

R. R. Coifman and G. Weiss, Analyse Harmonique Non-commutative sur Certains Espaces Homogènes (French) [Non-commutative harmonic analysis on certain homogeneous spaces] Lecture Notes in Math., 242, Springer, Berlin, 1971.

[14]

L. Diening, Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces, Bull. Sci. Math., 129 (2005), 657–700. doi: 10.1016/j.bulsci.2003.10.003.

[15]

Y. Ding and S. Lan, Anisotropic weak Hardy spaces and interpolation theorems, Sci. China Math., 51 (2008), 1690-1704. doi: 10.1007/s11425-008-0009-z.

[16]

Y. Ding and S. Lan, Anisotropic Hardy estimates for multilinear operators, Adv. Math., 38 (2009), 168-184.

[17]

Y. Ding and S. Lu, Hardy spaces estimates for multilinear operators with homogeneous kernels, Nagoya Math. J., 170 (2003), 117–133. doi: 10.1017/S0027763000008552.

[18]

Y. DingS. Lu and S. Shao, Integral operators with variable kernels on weak Hardy spaces, J. Math. Anal. Appl., 317 (2006), 127-135. doi: 10.1016/j.jmaa.2005.10.085.

[19]

R. Fefferman and F. Soria, The spaces weak H1, Studia Math., 85 (1986), 1-16. doi: 10.4064/sm-85-1-1-16.

[20]

J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam, 1985.

[21]

L. Grafakos, Hardy space estimates for multilinear operators, Ⅱ, Rev. Mat. Iberoam., 8 (1992), 69-92. doi: 10.4171/RMI/117.

[22]

L. Grafakos, Modern Fourier Analysis, 2ed edition, Graduate Texts in Mathematics 250, Springer, New York, 2009. doi: 10.1007/978-0-387-09434-2.

[23]

S. Hou, D. Yang and S. Yang, Lusin area function and molecular characterizations of MusielakOrlicz Hardy spaces and their applications, Commun. Contemp. Math., 15 (2013), 1350029, 37 pp. doi: 10.1142/S0219199713500296.

[24]

R. Johnson and C. J. Neugebauer, Homeomorphisms preserving Ap, Rev. Mat. Iberoam., 3 (1987), 249-273. doi: 10.4171/RMI/50.

[25]

N. J. Kalton, Linear operators on Lp for 0 < p < 1, Trans. Amer. Math. Soc., 259 (1980), 319-355. doi: 10.2307/1998234.

[26]

L. D. Ky, Bilinear decompositions and commutators of singular integral operators, Trans. Amer. Math. Soc., 365 (2013), 2931-2958. doi: 10.1090/S0002-9947-2012-05727-8.

[27]

L. D. Ky, New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators, Integral Equations Operator Theory, 78 (2014), 115-150. doi: 10.1007/s00020-013-2111-z.

[28]

L. D. Ky, Bilinear decompositions for the product space $H^1_L\times BMO_L$, Math. Nachr., 287 (2014), 1288–1297. doi: 10.1002/mana.201200101.

[29]

B. LiX. Fan and D. Yang, Littlewood-Paley theory of anisotropic Hardy spaces of Musielak-Orlicz type, Taiwanese J. Math., 19 (2015), 279-314. doi: 10.11650/tjm.19.2015.4692.

[30]

B. LiX. FanZ. Fu and D. Yang, Molecular characterization of anisotropic Musielak-Orlicz Hardy spaces and their applications, Acta Math. Sin., 32 (2016), 1391-1414. doi: 10.1007/s10114-016-4741-y.

[31]

B. Li, R. Sun, M. Liao and B. Li, Littlewood-Paley characterizations of anisotropic weak Musielak-Orlicz Hardy spaces, Nagoya Math. J., 1–40. doi: 10.1017/nmj.2018.10.

[32]

B. Li, D. Yang and W. Yuan, Anisotropic Hardy spaces of Musielak-Orlicz type with applications to boundedness of sublinear operators, The Scientific World Journal, (2014), 19 pp.

[33]

Y. LiangJ. Huang and D. Yang, New real-variable characterizations of Musielak-Orlicz Hardy spaces, J. Math. Anal. Appl., 395 (2012), 413-428. doi: 10.1016/j.jmaa.2012.05.049.

[34]

Y. LiangD. Yang and R. Jiang, Weak Musielak-Orlicz Hardy spaces and applications, Math. Nachr., 289 (2016), 637-677. doi: 10.1002/mana.201500152.

[35]

H. Liu, The weak Hp spaces on homogeneous groups, in Harminic Analysis (Tianjin, 1998), Lecture Notes in Math. Vol., 1984 (Springer, Berlin, 1991), pp. 113–118. doi: 10.1007/BFb0087762.

[36]

J. Liu, F. Weisz, D. Yang and W. Yuan, Littlewood-Paley and finite atomic characterizations of anisotropic variable Hardy-Lorentz spaces and their applications, J. Fourier Anal. Appl., (2018). doi: 10.1007/s00041-018-9609-3.

[37]

J. LiuD. Yang and W. Yuan, Anisotropic Hardy-Lorentz spaces and their applications, Sci. China Math., 59 (2016), 1669-1720. doi: 10.1007/s11425-016-5157-y.

[38]

J. LiuD. Yang and W. Yuan, Anisotropic variable Hardy-Lorentz spaces and their real interpolation, J. Math. Anal. Appl., 456 (2017), 356-393. doi: 10.1016/j.jmaa.2017.07.003.

[39]

J. Musielak, Orlicz Spaces and Modular Spaces, Springer-Verlag, Berlin-NewYork, 1983. doi: 10.1007/BFb0072210.

[40]

W. Orlicz, Über eine gewisse Klasse von Raumen vom Typus B (German), Bull. Int. Acad. Pol. Ser. A, 8 (1932), 207-220.

[41]

T. Quek and D. Yang, Calderón-Zygmund-type operators on weighted weak Hardy spaces over ${{\mathbb R}^n}$, Acta Math. Sin., 16 (2000), 141-160. doi: 10.1007/s101149900022.

[42]

E. M. SteinM. Taibleson and G. Weiss, Weak type estimates for maximal operators on certain Hp classes, Rend. Circ. Mat. Palermo, 1 (1981), 81-97.

[43]

J.-O. Strömberg and A. Torchinsky, Weighted Hardy Spaces, Berlin-New York: Springer-Verlag, 1989. doi: 10.1007/BFb0091154.

[44]

M. H. Taibleson and G. Weiss, The molecular characterization of certain Hardy spaces, Asterisque, Mathematical Reviews, 77 (1980), 67–149.

[45]

L. Tang, The weighted weak local Hardy spaces, Rocky Mountain J. Math., 44 (2014), 297-315. doi: 10.1216/RMJ-2014-44-1-297.

[46]

H. Wang, Boundedness of several integral operators with bounded variable kernels on Hardy and weak Hardy spaces, Internat. J. Math., 24 (2013), 1350095, 22 pp. doi: 10.1142/S0129167X1350095X.

[47]

D. Yang and S. Yang, Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated to non-negative self-adjoint operators satisfying Gaussian estimates, Commun. Pure Appl. Anal., 15 (2016), 2135-2160. doi: 10.3934/cpaa.2016031.

[48]

D. Yang, Y. Liang and L. D. Ky, Real-Variable Theory of Musielak-Orlicz Hardy Spaces, Lecture Notes in Mathematics, 2182, Springer-Verlag, Cham, 2017. doi: 10.1007/978-3-319-54361-1.

[49]

H. ZhangC. Qi and B. Li, Anisotropic weak Hardy spaces of Musielak-Orlicz type and their applications, Front. Math. China, 12 (2017), 993-1022. doi: 10.1007/s11464-016-0546-7.

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